Arrow’s Theorem Proves No Voting System is Perfect

One of the central issues in the theory of voting is described by Arrow’s Impossibility Theorem, which states roughly that no reasonably consistent and fair voting system can result in sensible results.

Named after Nobel Prize-winning economist Kenneth Arrow, the theorem starts by establishing a set of reasonable conditions on voting -- that is, on the method of aggregating individuals’ preferences into group preferences.

These conditions can lead to nonsensical group decisions, or manifestly undemocratic decision-making. As political scientists Ken Shepsle and Mark Bonchek put it in their book, Analyzing Politics, “The group is either dominated by a single distinguished member or has intransitive preferences.” For this reason, the theorem is sometimes known as the “dictator theorem.”

Understanding Arrow’s Theorem starts with understanding what economists and political scientists mean by “intransitive preferences.”

Preferences are known as “transitive” if they can be put in a sensible order. For instance, if you like apples best, then oranges, then bananas least, that means you prefer apples to oranges and to bananas, and oranges to bananas. If instead your fruit preferences cannot be put in best-to-least order -- you prefer apples to oranges and oranges to bananas, but prefer bananas to apples -- your preferences are known as “cyclic” or “intransitive.”

Arrow was trying to create a voting system that was consistent, fair, and would lead to transitive group preferences over more than two options. But in trying to create such a voting system, he proved that this was impossible.

The conditions Arrow put on a consistent and fair voting system can be expressed as the following:

1. Each voter can have any set of rational preferences. This requirement is called “universal admissibility.”

2. If every voter prefers choice A to choice B, then the group prefers A to B. This is sometimes called the “unanimity” condition.

3. If every voter prefers A to B, then any change in preferences that does not affect this relationship must not affect the group preference for A over B. For example, if a set of historians unanimously decides that Abraham Lincoln was a better president than Chester A. Arthur, a changing opinion of Bill Clinton should not affect this decision. This more subtle requirement is called “independence from irrelevant alternatives.”

4. There are no dictators.

Arrow’s Theorem states that, when choosing between more than two options, it is impossible in general to implement these four conditions without creating cycling group preferences. More dramatically, demanding transitive group preferences and the first three conditions implies there will be a dictatorship.

The formal proof is a tedious proof-by-contradiction, but it is easy to illustrate the problems with a common system, plurality voting.

In the plurality method, individuals vote only for their favorite candidate, and the candidate with the most votes wins. The trouble is, the winner might have fewer than fifty-percent of the vote.

Consider the 1992 U.S. presidential election. Clinton won the election with about 43 percent of the popular vote. George H.W. Bush had about 38 percent of the vote, and Ross Perot had about 19 percent.

Now, for the sake of argument, suppose that all Perot voters would have picked Bush if Perot had not run for reelection. Then, by 57 to 43 percent, Bush would have won the election. Roughly speaking, this result violates the independence from irrelevant alternatives condition.

Similar problems exist in all other voting systems, so political scientists and others have worked to figure out which conditions might reasonably be relaxed in order to create a sensible voting procedure. Many researchers consider the unanimity and no-dictator conditions sacrosanct, so attention has focused on irrelevant alternatives and, more importantly, on how often a particular system runs into problems.

Plurality, for example, does not lead to intransitive preferences as often as one might think. Shepsle and Bonchek calculated that, in a three-voter, three-candidate election, only 12 preference arrangements out of 216 possible arrangements led to intransitive group preferences.

Some argue that other voting systems are less prone to problems like those arguably experienced in the 1992 presidential election. Instant runoff voting and Cambridge’s version of proportional representation eliminate low-ranked candidates (like Perot) and redistribute votes among remaining candidates.

The Borda count system, used in some sports ranking schemes, also asks voters to rank candidates. Instead of elimination, points are assigned according to rankings, and these are used to determine a winner.

Each method has advantages, but each is guaranteed to have the disadvantages -- the sometimes-paradoxical results -- required by Arrow’s Theorem. The practical question for policy makers and voters is which system manages to run in to its problems least often.